Rational Krylov is used to solve an eigenvalue problem that is nonlinear in the eigenvalue parameter. In each iteration, it gives an improved shifted and inverted Arnoldi factorization to a linearization of the original nonlinear problem.
The final eigensolution gives a latent pair of the nonlinear eigenproblem and a starting guess for the next latent pair. Purging of uninteresting directions reduces the size of the basis. Locking of already found latent vectors gives deflation. An inner iteration makes the residual orthogonal before the size of the basis is increased. One sparse LU factorization can be used to locate several latent pairs.
Results, taken from the thesis of Patrik Hager, are reported for three test examples coming from finite element approximations modelling viscously damped vibrating structures.